Simulation of chaos-assisted tunneling in a semiclassical regime on existing quantum computers

Abstract

We present a quantum algorithm which allows to simulate chaos-assisted tunneling in deep semiclassical regime on existing quantum computers. This opens new possibilities for investigation of macroscopic quantum tunneling and realization of semiclassical Schrödinger cat oscillations. Our numerical studies determine the decoherence rate induced by noisy gates for these oscillations and propose a suitable parameter regime for their experimental implementation.PACS numbers: 03.67. Lx, 05.45.Mt, 75.45.+j Since 1935 until recently, the metaphor of Schrödinger cat oscillations between life and death [1] was considered as a purely theoretical concept. However, during the last decade such oscillations in a quantum limit were observed for two states of a Rydberg atom in a quantum cavity [2] and an experimental evidence was presented for a quantum superposition of macroscopically distinct states in a superconducting quantum interference device (SQUID) [3]. Manifestations of macroscopic quantum tunneling (MQT) were also observed in magnetization experiments with spin-ten molecular magnets F e 8 and M n 12 [4,5]. In addition to their fundamental interest these experiments promise to provide important applications, e.g. for solid-state qubit realization [6] and information storage [7] based on the Grover algorithm [8].The regime discussed in [1] assumes that the quantum tunneling takes place for a semiclassical object with a regular dynamics in two symmetric regions of phase space. Recently, the investigations of quantum chaos led to an extension of this concept to the phenomenon of chaos-assisted tunneling between islands of regular integrable dynamics separated by a chaotic sea [9]. In this case, due to chaos the period T u of tunneling oscillations becomes very sensitive to the variation of system parameters and statistical description should be used to describe the average distribution of T u [9][10][11]. This unusual tunneling is strongly influenced by complex instanton orbits and scarring effects [10,11] and a chaos enhancement of tunneling rate by orders of magnitude may be reached by a small variation of parameters. The first direct experimental observations of the chaos-assisted tunneling have been realized recently with cold [12] and ultracold atoms from a Bose-Einstein condensate [13] thus opening new possibilities for investigation of this interesting process. However, quantum tunneling is a very sensitive phenomenon and experimental studies are complicated by the decoherence produced by the environment. Due to that theoretical investigations of decoherence effects on MQT were initiated from the very beginning [14,15] and are continued up to now [16,17].In this Letter, we show that quantum entanglement and quantum computer simulations [18] can be efficiently used to study quantum tunneling in deep semiclassical regime. We illustrate this on the example of a quantum symplectic map (double well map) which has a rich phase space structure with integrable islands surrounded by chaotic sea. Our algorithm has certain...

Quantitative model for the effective decoherence of a quantum computer with imperfect unitary operations

“…Grover's algorithm [12] was studied in [13] with the help of a phenomenological model of probability diffusion, but the dependence of the model parameters on the number of qubits and ǫ was not determined. Conclusions similar to those in [7] were reached also in [14], where again the test algorithm was Grover's one 1 , and in a number of articles about the simulation of quantum chaotic maps [9,15,16,17,18]. These results show that the timescale for the degradation of the fidelity of a q.c.…”

“…Up to now, this method follows exactly the procedure described in [16]. The set W does not contain duplicated elements, so that its cardinality is smaller or equal to p (but it is always positive).…”

“…The described fidelity decay model has been tested numerically on two algorithms simulating periodically driven quantum system: the sawtooth map [9] and the double-well map [16]. These systems are usually studied in the quantum chaos field, because in spite of their relative simplicity they show a large variety of physical phenomena, from dynamical localisation to quantum ergodicity, first studied on the kicked rotator model [23].…”

“…The theoretical results are checked against numerical simulations based on two specific quantum algorithms calculating the evolution of a periodically driven quantum system (see appendix A), the sawtooth map [9] and the double-well map [16]. The former is a purely unitary transformation, while the latter uses an ancilla and intermediate measurements.…”

Quantum chaos and random matrix theory for fidelity decay in quantum computations with static imperfections

“…[9,10]). Such systems are described by chaotic quantum maps and include the quantum baker map [11], the quantum kicked rotator [12], the quantum saw-tooth map [13] and the quantum double-well map [14]. For them a map iteration can be performed for N -size vector in O(n q 2 ) or O(n q 3 ) gates while a classical algorithm would need O(n q 2 nq ) operations.…”